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[2014]

Excerpt from A. Carsetti, “Life, Cognition and Metabiology”.
As is well known, in order to induce a program to evolve we have to introduce some forms of pressure. According to Chaitin's model (2010), the Busy Beaver function exactly embodies a type of pressure. It represents the simplest possible challenge to force our organisms to evolve and to manifest their creativity. Using the Busy Beaver function the fitness of a new program can be compared to the fitness of the previous program. If the new program appears as fitter it takes the place of the old program. The Busy Beaver function of N, BB(N), that is used in AIT is defined to be the largest positive integer that is produced by a program that is less than or equal to N bits in size. BB(N) grows faster than any computable function of N and is closely related to Turing's halting problem, because if BB(N) were computable, the halting problem would be solvable. As Chaitin (2010) remarks, doing well on the Busy Beaver problem can utilize an unlimited amount of mathematical creativity. For example, we can start with addition, then invent multiplication, then exponentiation, then hyper-exponentials, and use this to concisely name large integers: N + N → N x N → N^N → (N^N)^N → .... According to Chaitin (2010): "There are many possible choices for such an evolving software model: You can vary the computer programming language and therefore the software space, you can change the mutation model, and eventually you could also change the fitness measure. For a particular choice of language and probability distribution of mutations, and keeping the current fitness function, it is possible to show that in time of the order of 2^N the fitness will grow as BB(N), which grows faster than any computable function of N and shows that genuine creativity is taking place, for mechanically changing the organism can only yield fitness that grows as a computable function" (Chaitin, 2010, p. 11). Subsequently Chaitin (2013) was also able to show that in time of the order of N²⁺^ε the fitness will grows as BB(N). In order to keep evolution from stopping, we stipulate that there is a non-zero probability to go from any organism to any other organism: -log₂ of the probability of mutating from A to B defines an important concept, the mutation distance, which is measured in bits. The conceptual complexity H (M) of the mutation M is the size in bits of the program M. This is the key idea utilized by Chaitin (2013) in order to model Darwinian evolution mathematically. We need a sufficiently rich mathematical space to model the space of all possible designs for biological organisms. In Chaitin's opinion (2013), the only space that is sufficiently rich to do that is a software space, the space of all possible algorithms in a fixed programming language. Here we can find a correct mathematical image for the afore mentioned space concerning the regulatory logic underlying cellular life. Compared to this conceptual framework, a systematic choice even more manageable as regards the fitness function is represented by Chaitin's Ω number. Chaitin (2010) uses Ω_k to define an organism O_k and a mutation M_k at time-step k, as well as the fitness function Ⱳ. Namely, an organism is defined by means of the first N(k) binary digits ω_i of Ω_k. The mutation acts on the organism by trying to improve the lower bounds on Ω. These mutations represent challenging an organism to find a better lower bound of Ω which amounts to an ever increasing source of knowledge. In this way, according to Chaitin's model (2010), evolution and randomness appear strictly intertwined. Let us note, however, that the statement that chaos is deterministic randomness is even more precisely formulated if the laws of chaos are predicated on the basis of the traditional undecidability theory. For example, it is possible to demonstrate how in specific dynamical systems, chaos is simply the effect of the undecidability proper to a recursive, system-related algorithm. At an even more general level, we can state that deterministic, chaos-generating systems may be considered as set-ups able to act as models (in a logical-mathematical sense) for formal systems (algorithms) within which it is possible to trace the trajectory of a mathematically realized randomness. In other words, it is possible to consider the randomness defined at an algorithmic-informational level, as the counterpart and theoretical foundation of physical randomness connected to the articulation of well-defined processes – i.e. set-ups. This last kind of randomness, however, must be considered as absolute, not only because it satisfies the requisites of objectivity-coherence and intrinsic incompressibility, but also because the cognitive agent is able to define it principally by stating the limitation theorems concerning the formal system within which randomness is recognized as such, particularly with regard to that specific system which, at an algorithmic and programming level, assists, as we have just said, in the “guided” construction of the corresponding set-up. Indeed, when the cognitive agent becomes aware of the fact that a particular axiomatic system allows him to define a given number as random, but is still unable to resort to its formal tools in order to calculate that number or to permit a proof of its purely casual nature, s/he sees a concrete foreshadowing of an absolutely random string which, out of all possible messages, will be that which is most redolent of information precisely because all redundancy has been eliminated. Chaitin (2013) is perfectly right to bring the phenomenon of evolution in his natural place which is a place characterized in a mathematical sense: Nature "speaks" by means of mathematical forms. Life is born from a compromise between creativity and meaning, on the one hand, and, on the other hand, is carried out along the ridges of a specific canalization process that develops in accordance with computational schemes. It is with respect to this particular channeling that the fitness must be calculated according to the deployment of precise functions. Hence the emergence of those particular forms (linked to specific development languages) that are instantiated, for example, by the Fibonacci numbers, by the fractal-like structures etc. that are ubiquitous in Nature. As observers we see these forms but they are at the same time inside us, they pave of themselves our very organs of cognition. In disagreement with Kant, we have to recognize that, at the level of a biological cognitive system, sensibility is not a simple interface between absolute chance and an invariant intellectual order. On the contrary, the reference procedures, if successful, are able to modulate canalization and create the basis for the appearance of ever-new frames of incompressibility through morphogenesis. This is not a question of discovering and directly exploring (according, for instance, to Putnam’s conception) new “territories”, but of offering ourselves as the matrix and arch through which they can spring autonomously in accordance with ever increasing levels of complexity. There is no casual autonomous process already in existence, and no possible selection and synthesis activity via a possible “remnant” through reference procedures considered as a form of simple regimentation. These procedures are in actual fact functional to the construction and irruption of new incompressibility: meaning, as Forma formans, offers the possibility of creating a holistic anchorage, and is exactly what allows the categorial apparatus to emerge and act according to a coherent “arborization”. The new invention, which is born then shapes and opens the (new) eyes of the mind: I see as a mind because new meaning is able to articulate and take root through me . In other words, at the biological level, what is innate is the result of an evolutionary process and is “programmed” by natural selection. Natural selection is the coder (once linked to the emergence of meaning): at the same time at the biological level this emergence process is indissolubly correlated to the continuous construction of new formats in accordance with the unfolding of ever new mathematics, a mathematics that necessarily moulds coder’s activity. Hence the necessity of articulating and inventing a mathematics capable of engraving itself in an evolutionary landscape in accordance with the opening up of meaning. In this sense, for instance, the realms of non standard-models and non-standard analysis represent, today, a fruitful perspective in order to point out, in mathematical terms, some of the basic concepts concerning the articulation of an adequate intentional information theory. This individuation, on the other side, presents itself not only as an important theoretical achievement but also as one of the essential bases of our very evolution as intelligent organisms. ( Read more: A. Carsetti, “Life, Cognition and Metabiology”, Cognitive Processing, 15 (4) (2014) : 423-434).
[2013 - 2015]

Excerpt from A. Carsetti, "Intended Models and Henkin Semantics".
The mathematical practice considered as a capacity that passes through us and that comes to live within us determining what we really manage to become as observers living at the natural level and permitting us to reach true existence also manages, hand in hand, to reveal some of its mysterious recesses affording a glimpse of the deep dialectic underlying it. The fundamental problem is that we have to interrogate ourselves positing ourselves as oracles: Noli foras ire. In te ipsum redi. The oracle and the inspirited truth can come to express themselves only in us and through us. We are within a path where the key is represented by the real possibility to change minds (and semantics) but in accordance with the transformation of our own fibers. Thus, we are no longer faced with intuitions concerning numbers as objects, their formation processes and their use: the numbers are natural because they arise from the progressive unfolding of specific primordial capacities. To the extent that these capacities embody according to specific cognitive processes they give rise to numbers, presiding over their formation in a natural way. Everything, however, is also linked to the previous identification of a Method: the “irruption” and the development of the original capacities is to the extent of the interrogation of the oracle and the exploration of the non-standard realm. It is in this sense that, starting from the unfolding of a specific set of capacities (at the level of the original “magma”), counting appears closely related to the establishment of a recursive ordering. but if we look at this kind of process in a more accurate way, we can discover that counting is also related to self-reference and to the progressive unfolding of a tissue of eigenforms. Counting presupposes: a) the full articulation of a set of capacities; b) the actual existence of specific control procedures able, first of all, to manage recursions (in the light of an ongoing development of the software of meaning); c) a process of mutual sharing determining the rise of specific forms of rational perception concerning the effective existence of real objects (the numbers) with respect to a set of observers. As we have seen, in the light of Set Theory and ZF, it is very easy to show that the construction of numbers is a recursive process. What is less obvious or less known is the role played by the fixed points at the level of that fundamental process that, at the biological level, undermine the construction of numbers in an operative way. L. Lawvere in 1970 found a new way to prove a fundamental theorem by Georg Cantor which is at the basis of the development of Set Theory and the successive outlining of ZF. As is well known, Cantor’s theorem affirms that for every set S, there are more subsets of S than there are members of S. Today the right way to prove this theorem is first to prove Lawvere’s fixed point theorem (Lawvere, 1969):
1.8. Theorem (Lawvere)
Suppose e : A→B^A is a surjective map. Then every map f : B→B has a fixed point, i.e., x ∈ B such that f (x) = x. Proof. Consider the map s: A→B defined by s (x) = f (e (x) (x)). Because e is onto, there is y ∈ A such that e (y) = s. We have e (y)( y) = s (y) = f (e (y)( y)), therefore e (y)( y) is a fixed point of f. QED.
1.9. Corollary (Cantor)
There is no onto map from a set A onto its powerset 𝓟(A). Proof. 𝓟(A) is isomorphic to 2^A where 2 = {0, 1}. In this sense, the subsets of A correspond to their characteristic functions A→ {0, 1}. Suppose e : A→ 2^A is onto. But by Lawvere’s theorem every map 2 →2 has a fixed point. On the contrary, this is not the case for the map f (x) = 1- x. Therefore no such e can exist. QED.
We have just seen, by utilizing the Compactness theorem, that there is a countable non-standard model of arithmetic. on the other hand, we have also seen that, on the basis of a theorem by Henkin, the order type of any non-standard model of arithmetic is of the form ω + (ω* + ω )· η where ω is the order type of the standard natural numbers, ω* is the dual order (an infinite decreasing sequence) and η is the order type of the rational numbers. A countable non-standard model begins with an infinite increasing sequence (the standard element) that is followed by a collection of “blocks”, each of order type (ω* + ω ), the order type of the integers. These blocks are in turn densely ordered with the order type of the rationals. Actually, the non-standard numbers must be dense and linearly ordered without endpoints. Nevertheless, it follows from compactness that this entire non-standard structure satisfies all the axioms of Peano arithmetic; i.e., that all the arithmetical theorems are true in this structure. In other words, we have that any countable non-standard model of arithmetic has order type ℕ+ℤℚ . Actually, up to isomorphism, the only countable dense linear order without endpoints is ℚ. However, if the order type of countable non-standard models is well known, the correlated arithmetical operations appear much more sophisticated. As a matter of fact, the arithmetical structure differs from ω + ( ω* + ω ) ∙ η . Actually, as we have just seen, Tennenbaum’s theorem shows that for any countable non-standard model of Peano arithmetic there is no way to code the elements of the model as standard natural numbers such that either the addition or multiplication operation of the model is computable on the codes. In accordance with this line of research, we are faced, now, with some important theoretical developments. As an example, let us introduce the following theorem:
1.10. Theorem (Klaus Potthoff)
There is no non-standard model of arithmetic with order type ℕ+ℤℝ .
These developments open the way to some interesting applications of non-standard models to non-standard Analysis. The idea, for instance, that infinitesimals can be reconstructed utilizing non-standard natural numbers was introduced by A. Robinson. Actually, a non-standard model of Peano arithmetic can easily be extended to an ordered field. If x is a non-standard number, the number 1/x is non-zero but infinitely small. Hence the possibility to reconstruct in a rigorous way the ancient idea of infinitesimals resorting to non-standard numbers. In order to better understand the link between Skolem results, the inner structure of reflexive domains and non-standard analysis, let us resort to an exemplification: the von Koch curve is an eigenform, but it is also a fractal, however, it can also be designed utilizing the sophisticated mechanisms of non-standard analysis. In this last case, we have the possibility to enter a universe of replication, which also opens to the reasons of real emergence. At this level, the growth of the linguistic domain, the correlated introduction of ever-new individuals appear strictly linked to the opening up of the software of meaning and to a continuous unfolding of specific emergence processes with respect to this very opening. Hence the need for the introduction of precise evolutionary parameters, the very necessity, in general, to bring back the inner articulation of the living eigenforms not only to the structures of a“simple” perceptual activity (as linked to the mere identification of the fixed points at stake) but also to the structures connected to the canalization processes concerning intentionality in action. Actually, our perceptual activity is conditioned by the unfolding of the embodiment process and is linked to the cues offered by meaning to the ongoing reflexive procedures. A new world opens up before (and within) our eyes whenever we can grasp the meaning of things. Each time the ribs of this world appear related to a continuous development of the mathematical language in action as linked to an effective exploration of the non-standard realm. Hence the very possibility of the continuous emergence of an unheard intentionality that will come to present itself only in accordance to a radical metamorphosis: it is in my own firewood in the bush that the instrument forged over the course of a life can come to find new lymph but in the Other and in my own overcoming. (Read more: A. Carsetti, "Intended Models and Henkin Semantics", La Nuova Critica, vol. 61-62 (2013) : 125-180).
[2016]

Excerpt from A. Carsetti, "Towards a Conceptual Metabiology" (to appear). (TBI)